In a right triangle (right-angled triangle) the longest side is always furthest away from the right angle. It is called the hypotenuse. The length of the hypotenuse can be calculated from the lengths of the two other sides. In the diagram, c is the hypotenuse and we can calculate it from a and b.

This is a statement of the Pythagorean Theorem or Pythagoras' Theorem - as an equation relating the lengths of the sides a, b and c:[1]

a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}}

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

This only works for right triangles!

For the right angle triangle with corners labelled by A,B,C as shown, the following says the same thing:

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

We can also show this equation with a diagram, as on the right, where each side of a right angle triangle has a square attached to it. This is the geometrical interpretation of the Pythagorean theorem, looking at the theorem as a theorem about areas. The areas of the smaller squares add up to the area of the larger square.

In terms of areas, the Pythagorean Theorem states:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

The two ways of talking about the Pythagorean theorem amount to exactly the same thing, since the area of a square is just the length of one side squared.

The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is much evidence that Babylonian mathematicians understood the formula.[4]

Another often used example is a right triangle in which the two remaining angles are 45∘{\displaystyle 45^{\circ }} and both a=1{\displaystyle a=1} and b=1{\displaystyle b=1} . This time the answer doesn't come out as a whole number:

One more example a right triangle in which the shorter sides are p=11{\displaystyle p=11} cm and q=7{\displaystyle q=7} cm.

What's this about p{\displaystyle p} and q{\displaystyle q} ? It really doesn't matter what letter we use to name the lengths of the sides as long as we are consistent. The formula in the blue box still holds. In this case we could call the hypotenuse r{\displaystyle r} , and so we have p2+q2=r2{\displaystyle p^{2}+q^{2}=r^{2}} .

It's worth checking the answer makes some kind of sense. If the short sides are 11 cm and 7 cm is it reasonable that the long side is about 13 cm? Yes, it is. If you'd got an answer of 20 cm (or 170 cm) it would be unreasonable. The two other sides won't stretch that far. Or if the answer were 11 cm or less than you'd know that you hadn't got the length of the hypotenuse, the longest side. We're OK here with this answer of 13.04 cm, which we probably should round down to 13 cm.

This example is in three dimensions and again we have to use the Pythagorean Theorem twice.

The great pyramid of Khafre is 274 cubits high. It has a square base and the square has sides of length 412 cubits. What is the length of the sloping diagonal edge from a corner up to the summit?

To answer this question we work in two stages. See the diagram.

The line AO¯{\displaystyle {\overline {AO}}} is vertical and the line OB¯{\displaystyle {\overline {OB}}} is horizontal, so there is a right angle triangle with sides AO¯{\displaystyle {\overline {AO}}} and OB¯{\displaystyle {\overline {OB}}} . This lets us calculate the length AB¯{\displaystyle {\overline {AB}}} .

We need to be careful. AO¯{\displaystyle {\overline {AO}}} is 274 cubits, but OB¯{\displaystyle {\overline {OB}}} is half the length of the side of the square, so OB¯{\displaystyle {\overline {OB}}} is 4122=206{\displaystyle {\tfrac {412}{2}}=206} cubits.

What is the length of the hypotenuse AB¯{\displaystyle {\overline {AB}}} of the triangle △AOB{\displaystyle \triangle AOB}? Well,

We've made progress, but we want the length AC¯{\displaystyle {\overline {AC}}} , not the length AB¯{\displaystyle {\overline {AB}}} .

Now we have another right angle triangle, △ABC{\displaystyle \triangle ABC} with the right angle at B. We want the length AC¯{\displaystyle {\overline {AC}}} . We have just calculated AB¯{\displaystyle {\overline {AB}}} and we know, because the base of the pyramid is a square, that BC¯{\displaystyle {\overline {BC}}} is once again 206 cubits. This time AC¯{\displaystyle {\overline {AC}}} is the hypotenuse (of △ABC{\displaystyle \triangle ABC}). So

A cubical box is used to ship a printer. If one side is 90 cm, what is the longest distance between any two corners of the box?

An LCD television screen is measured 26 inches from corner to corner. The screen height is 13 inches. How wide is the screen? (To one-tenth of an inch)

What is the length of the longest line in this diagram?

Pythagoras again and again

In this problem you can assume that all the angles that look like right angles are.

Aliens from the fourth dimension are packing their holographic TV into a box ready for a trip to Earth. The box measures 1.5 Angstroms by 1.5 Angstroms by 1.5 Angstroms by 1.5 Angstroms (it is a 4 dimensional hypercube). What is the length of the longest diagonal?

If a room is 17 ft long, 14 ft wide, and 10 ft high, what is the length of the diagonal of (a) the floor, (b) an end wall, and (c) a side wall of the room?

Find the missing side (Pythagorean triples)

In each of these examples you are told that

All three sides of the triangle are whole numbers.

The triangle is a right angle triangle

You are given two of the sides.

Find the third side. Which side is the hypotenuse?

Sides you are told are 4,5, what is the third side?

Sides you are told are 5,12, what is the third side?

Sides you are told are 6,8, what is the third side?

Sides you are told are 40,41, what is the third side?

Squaring small numbers

What is (−0.5)2{\displaystyle (-0.5)^{2}} ?

Of the numbers between -2 and +2, which ones end up closer to zero when you square them?